Differential Geometry of Singular Spaces

奇异空间的微分几何

• 批准号: 5407091
• 负责人: Professor Dr. Andreas Bernig
• 金额: --
• 依托单位: Schwerpunkt Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2005-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5407091?language=en
• 关键词: Differential; Geometry; Singular; Spaces

The study of Riemannian manifolds is central in mathematics. They are smooth objects which can be analyzed using geometric and analytic tools. It turned out over the last fifteen years that the study of certain limits of Riemannian manifolds is very fruitful. In general, these limits present difficult singularities. Examples for limit spaces are metric spaces with a lower or an upper curvature bound. In this project, we want to use Geometric Measure Theory in order to construct and study other types of limit spaces which are better suited for questions about scalar curvature and Ricci curvature. There will be an interesting interplay between Differential Geometry, Geometric Measure Theory, Convex Geometry and Subanalytic Geometry. Hopefully, there will be applications of this theory to the existence of Einstein metrics, realization of Yamable invariants and to the geometry of Riemannian manifolds with scalar or Ricci curvature bounds.

黎曼流形的研究是数学的中心。它们是光滑的物体，可以使用几何和分析工具进行分析。事实证明，在过去的十五年里，研究某些限制的黎曼流形是非常富有成果的。一般来说，这些极限会出现困难的奇点。极限空间的例子是具有曲率下界或曲率上界的度量空间。在这个项目中，我们希望使用几何测度理论来构造和研究其他类型的极限空间，这些空间更适合于有关纯量曲率和里奇曲率的问题。微分几何、几何测度论、凸几何和次解析几何之间将有有趣的相互作用。希望这个理论能应用于爱因斯坦度量的存在性、Yamable不变量的实现以及具有标量或Ricci曲率边界的黎曼流形的几何。